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s p a tio t e m po r al disp e r sion. II. S a t u r a ble sys t e m s

Ch ris ti a n, JM, McDo n ald, GS, Lun die, MJ a n d Kots a m p a s e ris, A

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Tit l e Rela tivis tic a n d p s e u do r el a tivis tic for m ula tion of no nline a r e nvelope e q u a tions wi th s p a tio t e m po r al dis p e r sion. II. S a t u r a ble sys t e m s

Aut h or s Ch ris ti a n, JM, M cDon ald, GS, Lun die, MJ a n d Kots a m p a s e ris , A

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Relativistic and pseudorelativistic formulation of nonlinear envelope equationswith spatiotemporal dispersion. II. Saturable systems

J. M. Christian,∗ G. S. McDonald, M. J. Lundie, and A. KotsampaserisJoule Physics Laboratory, School of Computing, Science and Engineering,University of Salford, Greater Manchester M5 4WT, United Kingdom

(Dated: July 26, 2018)

We consider an envelope equation with space-time symmetry for describing scalar waves in systemswith spatiotemporal dispersion and a generic saturable nonlinearity. Exact bright and gray solitonsare derived by direct integration methods and coordinate transformations, with the results for cubic-quintic systems [see companion article—submitted to Phys. Rev. A] recovered in the limit of weaksaturation. Classic predictions from a nonlinear-Schrodinger formulation of the propagation problemare shown to emerge asymptotically as subsets of the more general spatiotemporal solutions. Therobustness of the new solitons against perturbations to the local pulse shape is then tested bydeploying integral stability criteria, symmetry principles, and numerical analysis.

PACS numbers: 42.65.-k, 42.65.Fs, 42.65.Tg, 42.65.Wi, 05.45.YvKeywords: bright solitons, dark solitons, Kerr effect, spatial dispersion, waveguide optics

I. INTRODUCTION

Solitons and solitary-wave phenomena are elementaryexcitations, pervasive throughout Nature, that play a keyrole in modern understandings of nonlinear systems [1].Regardless of the physical context—optical or fluidic, me-chanical or electromagnetic, biological or chemical, classi-cal or quantum—the emergence of these robust particle-like wave states generally requires just two basic ingredi-ents: linear dispersion and nonlinearity [2]. Models sup-porting envelope solitons are historically rooted in uni-versal equations such as the nonlinear-Schrodinger (NLS)type, and in the simplest case they describe waveformsthat are localized in time t and travelling through space z.Longitudinal modulations to a rapidly-varying harmonicsignal are typically assumed to take place on a scale-length that is much greater than the carrier wavelength[the slowly-varying envelope approximation (SVEA)].

The SVEA is often a valid starting point for analy-ses [3] and the undoubted success of NLS-based modelsin predicting experimentally-observed phenomena (e.g.,in optics [4]) has allowed them to become linchpins ofconventional pulse theory [5]. The SVEA is usually com-plemented by a Galilean transformation, where for con-venience one boosts from the laboratory frame to a local-time frame moving at the group velocity vg, as definedby coordinates zloc ≡ z and tloc ≡ t − z/vg. It is in-structive, however, to pose questions about the prop-erties of wavepackets beyond the more traditional levelof description. What are the governing equations? Dothey have exact non-trivial solutions? Can invariancelaws and conserved quantities be identified? What arethe implications for stability? Can predictions be recon-ciled with the SVEA? etc. We have made some progressin answering these fundamental questions by proposing

∗ Corresponding author: j.christian@salford.ac.uk

a spatiotemporal formulation based on generic envelopeequations that respect space-time symmetry [6].

The spatiotemporal nomenclature has different inter-pretations so it is helpful to begin by clarifying the ter-minology used throughout (the situation is akin to non-paraxial when discussing beams in spatial optics [7]). It isoften associated with the phenomenon of wave-like “bul-lets” where the interplay between nonlinearity, group-velocity dispersion (GVD) and diffraction can sustain (atleast in the short term) wave self-confinement in trans-verse and longitudinal spatial directions in addition tobeing localized in time [8]. One also encounters theterm in Ginzburg-Landau contexts when analyzing pat-tern emergence, bifurcations, and chaotic dynamics un-der the combined action of gain and loss [9]. Here, wetake it to mean the simultaneous presence of both spatialand temporal dispersion in any wave-based system [6].

Mathematically, our quite general conception of spa-tiotemporal dispersion lies with the appearance of both∂2/∂z2 and ∂2/∂t2 operators in the envelope equation(which may or may not also involve formal diffractiveconsiderations). It has application in the field of non-linear optics, supplementing the widely-known SVEA-based models of light propagation [3–5] with a more com-plete geometrical theory [6]. It also has practical usesin terms of modelling condensed-matter effects in somespecial classes of semiconductor (e.g., ZnCdSe/ZnSe su-perlattices). Bianacalana and Creatore [10] have previ-ously demonstrated that material spatial dispersion dueto photon-exciton coupling cannot be captured by SVEA-type frameworks, and that retaining full generality of∂2/∂z2 facilitates an adequate description.

To date, the classic cubic [11] and cubic-quintic (seecompanion article [12]) nonlinearities have been inves-tigated in some detail with particular emphasis placedon deriving exact bright and dark solitons. One of themost self-evident and intriguing results is that our spa-tiotemporal formulation has strong analogues with spe-cial relativity [13]. The invariance laws are either rela-

2

tivistic or pseudorelativistic (depending on the interplaybetween spatial and temporal dispersion), and as a con-sequence (inverse) velocities add geometrically througha rule that is reminiscent of the familiar Lorentz form.Perhaps most striking is the recovery of SVEA-type re-sults. They emerge asymptotically through a multiple-limit procedure that is entirely equivalent to the wayNewtonian physics reappears from special relativity atlow speeds. For completeness, there remains one finalclass of system nonlinearity that needs to be addressed.

Saturation is a universal phenomenon that tends to in-hibit unphysical run-away effects. In optics, for example,it is a principal feature of Maxwell-Bloch cavity modelsinvolving the continuous interaction between a circulat-ing light beam and a system comprising atoms with twodiscrete energy levels [14]. Such a system tends to exhibitsaturation in its medium polarization and population in-version when electric-dipole coupling is subjected to high-intensity pumping. In cavityless configurations, such asa dispersive waveguide, one might intuitively expect thenonlinear dielectric response (e.g., permittivity or refrac-tive index) of the core material [15] to support a maxi-mum allowable change before approaching the thresholdfor optical damage. However, the standard cubic (i.e.,Kerr) [3, 16] and cubic-quintic [17, 18] approximationspermit arbitrarily-large contributions to self-interactioneffects. These models are based on simple scalar powerseries (or, more rigorously, on tensor expansions of themedium polarization and the introduction of nonlinearsusceptibilities [16, 19]) that are truncated after a finitenumber of terms (e.g., on the basis of order-of-magnitudeconsiderations). In contrast, the plateauing that char-acterizes saturation (i.e., where a physical property cansustain no further self-induced variation) does not alwayslend itself well to polynomial-type representations that,inevitably, fail at sufficiently high light intensity.

The saturating responses of many optical materialshave been measured over the years. Such experimentalstudies have included some semiconductor-doped glasses(e.g., CdSSe and Schott OG 550 glass) [20], ion-dopedcrystals (e.g., GdAlO3:Cr3+) [21], bio-optical media [22],π-conjugated polymers [23], and various photorefractivecrystals (e.g., LiNbO3 and SBN) [24].

While several trial functions are available for describ-ing a saturable refractive index [25–27], all of whichshare similar qualitative features, our principal inter-est lies with that proposed by Wood, Evans, and Ke-nan [28]. Their model appears to be unique in thatit allows the corresponding governing equations to beintegrated exactly—for instance, families of transverseguided modes in dielectric planar waveguides were ob-tained by solving the Helmholtz equation and enforc-ing continuity conditions at the boundary between sub-strates. More pertinently, exact bright [29] and dark [30]solitons derived by Krolikowski and Luther-Davies arealso known within the context of Schrodinger equations.It is these latter solution classes that we seek to gener-alize here and, in so-doing, more fully develop our un-

derstanding of waves in space-time-symmetric systemsby accounting for saturation effects. Known spatiotem-poral cubic-quintic [12] and cubic [11] solitons are thenexpected to emerge as special cases, forming a naturalhierarchy of solutions.

The layout of this paper is as follows. In Sec. II, we ex-plore the spatiotemporal envelope equation in the contextof the saturable nonlinearity and formulate the intensity-phase quadrature problem. Exact bright and dark soli-tons are derived in Secs. III and IV, respectively, withcoordinate transformations deployed in Sec. V to ob-tain more general results (including a discussion of non-degenerate bistability characteristics). Rigorous asymp-totic analyses are detailed in Sec. VI with regards variousimportant physical limits, with predictions about soli-ton stability made and tested against full simulations inSec. VII. We conclude, in Sec. VIII, with some remarksabout connections to other potential research avenues.

II. SPATIOTEMPORAL MODEL

A. Envelope equation

We consider the governing equation for a dimensionlessenvelope u that is given by

κ∂2u

∂ζ2+i

(∂u

∂ζ+ α

∂u

∂τ

)+s

2

∂2u

∂τ2

+1

2

2 + |u|2/ρsat

(1 + |u|2/ρsat)2 |u|

2u = 0, (1)

where ζ and τ are normalized space and time coordi-nates, respectively, as measured in the laboratory frame.The linear part of the wave operator in Eq. (1) is generic(in terms of its dispersive contributions and space-timesymmetry) while the nonlinear part is homogeneous andparametrized by ρsat (the normalized saturation inten-sity). We assume a set of units such that GVD is con-trolled by s = ±1 (+1 for anomalous, −1 for normal) andthe parameter α is a ratio of group velocities. Spatial dis-persion is determined by κ O(1), and it is taken to bepositive without loss of generality. An example scaling isgiven in Appendix A.

To clarify, here we define relativistic and pseudorela-tivistic scenarios as being those characterized by s = −1and s = +1, respectively, whereupon the transformationlaws of Eq. (1) correspond to skews and rotations in the(τ, ζ) plane [11].

In the context of Eq. (1), the SVEA is embodied by theinequality κ|∂2u/∂ζ2| |∂u/∂ζ|. Throughout, we pur-posely avoid that regime (both analytically and compu-tationally) until considering the position of conventionalmodel predictions within the wider soliton hierarchy. Ne-glecting the operator κ∂2/∂ζ2 will then be shown as syn-onymous with a simultaneous multiple limit in the alge-bra of spatiotemporal solutions. Moreover, we have found

3

that κ∂2/∂ζ2 can be accommodated in many exact analy-ses and thus it does not need to be treated perturbatively(e.g., through order-of-magnitude considerations [31]).

B. General quadrature equations

Solutions to Eq. (1) are sought that have a generalform described by the Madelung-type ansatz

u(τ, ζ) = ρ1/2(τ, ζ) exp[iψ(τ, ζ)], (2a)

where ρ and ψ are real functions determining the inten-sity and (total) phase, respectively, of u. SubstitutingEq. (2a) into Eq. (1) and collecting the real and imagi-nary parts yields the following pair of coupled equations:

2

ρ

(∂2ρ

∂τ2+ 2sκ

∂2ρ

∂ζ2

)− 1

ρ2

[(∂ρ

∂τ

)2

+ 2sκ

(∂ρ

∂ζ

)2]

− 4

[(∂ψ

∂τ

)2

+ 2sκ

(∂ψ

∂ζ

)2]− 8s

(∂ψ

∂ζ+ α

∂ψ

∂τ

)+ 8s

2 + ρ/ρsat

(1 + ρ/ρsat)2

(ρ2

)= 0, (2b)

ρ

(∂2ψ

∂τ2+ 2sκ

∂2ψ

∂ζ2

)+

(∂ψ

∂τ

∂ρ

∂τ+ 2sκ

∂ψ

∂ζ

∂ρ

∂ζ

)+ s

(∂ρ

∂ζ+ α

∂ρ

∂τ

)= 0 (2c)

(neglecting all the κ-dependent terms leads to the well-known fluid-type equations [32] for ρ and ψ in the cor-responding SVEA-type model). One can now eliminatethe carrier-wave part of the solution by setting

ψ(τ, ζ) = Ψ(τ, ζ) +Kζ − ζ

2κ. (3)

Here, Ψ denotes the phase distribution for the solitaryexcitation, K is the propagation constant, and the finalfactor describes the rapidly-oscillating component inher-ent to the solutions of models that are second-order in ζ[11]. Quadrature equations (2b) and (2c) may then bewritten as

2

ρ

(∂2ρ

∂τ2+ 2sκ

∂2ρ

∂ζ2

)− 1

ρ2

[(∂ρ

∂τ

)2

+ 2sκ

(∂ρ

∂ζ

)2]

− 4

[(∂Ψ

∂τ

)2

+ 2sκ

(∂Ψ

∂ζ

)2]

− 8s

(α∂Ψ

∂τ+ 2κK

∂Ψ

∂ζ

)− 8s

[β − 2 + ρ/ρsat

(1 + ρ/ρsat)2

(ρ2

)]= 0 (4a)

and

ρ

(∂2Ψ

∂τ2+ 2sκ

∂2Ψ

∂ζ2

)+

(∂Ψ

∂τ

∂ρ

∂τ+ 2sκ

∂Ψ

∂ζ

∂ρ

∂ζ

)+ s

(α∂ρ

∂τ+ 2κK

∂ρ

∂ζ

)= 0. (4b)

Here, we have defined the dispersion relation by identi-fying κK2 − 1/4κ ≡ β so that

K = ± 1

2κ

√1 + 4κβ, (4c)

and where the ± sign denotes propagation in the forward(+) or backward (−) longitudinal direction.

C. Space-time coordinate transformation& symmetry reduction

Inspection of Eqs. (4a) and (4b) demonstrates thatthere is a symmetry between space and time derivativesthat does not appear in the conventional approach topulse modelling. Such symmetry can be exploited by in-troducing the lumped space-time coordinate ξ ≡ ξ(τ, ζ),effectively a single new independent variable, where

ξ(τ, ζ) ≡ τ − V0ζ√1 + 2sκV 2

0

(5)

and V0 is a constant intrinsic velocity that parametrizesthe transformation. By implementing change-of-variables (5) through the replacement of partial differen-tial operators ∂/∂τ and ∂/∂ζ and combinations thereof—see Ref. [12] for details—we can use Eqs. (4a) and (4b) towrite down a pair of coupled ordinary differential equa-tions for the ρ and Ψ quadratures in terms of ξ:

d

dρ

[1

ρ

(dρ

dξ

)2]− 4

(dΨ

dξ

)2

− 8s

(α− 2κKV0√

1 + 2sκV 20

)dΨ

dξ

− 8s

[β − 2 + ρ/ρsat

(1 + ρ/ρsat)2

(ρ2

)]= 0, (6a)

ρd2Ψ

dξ2+dΨ

dξ

dρ

dξ+ s

(α− 2κKV0√

1 + 2sκV 20

)dρ

dξ= 0. (6b)

To find particular (e.g., soliton) solutions, Eqs. (6a) and(6b) must be supplemented by appropriate boundaryconditions on ρ(ξ) and Ψ(ξ).

III. BRIGHT SOLITON PULSES

One expects bright solitons to exist in the case ofanomalous GVD (where s = +1). In the following anal-ysis, the ‘b’ subscript is introduced to denote bright soli-tons through ρ(ξ)→ ρb(ξ), Ψ(ξ)→ Ψb(ξ), β → βb, K →Kb, and the intrinsic velocity is flagged by V0 → V0b.

4

A. Intensity quadrature

To obtain the quadrature equations for bright solitons,we look for particular solutions where Ψb = 0 so thatthere is no phase change across the temporal width ofthe pulse. Equation (6a) becomes

d

dρb

[1

ρb

(dρb

dξ

)2]

= 8

[βb −

2 + ρb/ρsat

(1 + ρb/ρsat)2

(ρb

2

)],

(7a)

and direct integration with respect to ρb leads to(dρb

dξ

)2

= 8βbρ2b − 4ρ2

sat

(ρ2

b

ρsat+

ρb

1 + ρb/ρsat

)+ c2bρb,

(7b)where c2b is a constant to be determined from the solu-tion boundary conditions. As ξ → ±∞, the intensityprofile must decay to zero sufficiently rapidly so thatρb → 0 as (dρb/dξ)

2 → 0. Applying these limits toEq. (7b) shows that c2b = 4ρ2

sat (note that, in the cubic-quintic regime [12], the corresponding constant of inte-gration vanishes). Similarly, when ξ → 0 the intensitytends to its peak value, denoted by ρ0, so that

βb =1

1 + ρ0/ρsat

(ρ0

2

). (7c)

The bright soliton propagation constant Kb, given inEq. (4c), is then fully determined.

By combining the algebraic results for c2b and βb, theright-hand side of Eq. (7b) can be factorized to yield themore compact structure(

dρb

dξ

)2

= 4ρ2b

(ρ0 − ρb)

(1 + ρ0/ρsat) (1 + ρb/ρsat). (8)

A second integration uncovers an implicit result describ-ing the spatiotemporal pulse intensity profile which, forease of comparison, we express in a form similar to thatintroduced by Krolikowski and Luther-Davies [29]:

tan−1 (ηb) +1

2

(ρsat

ρ0

)1/2

ln

[(ρ0/ρsat)

1/2 + ηb

(ρ0/ρsat)1/2 − ηb

]=

ρ1/2sat

(1 + ρ0/ρsat)1/2

ξ, (9a)

where

ηb[ρb] ≡(ρ0 − ρb

ρsat + ρb

)1/2

(9b)

is a positive real parameter and ηb(ξ) ≡ ηb[ρb(ξ)]. Equa-tions (9a) and (9b) can then be solved numerically toobtain ρb(ξ) (see Fig. 1). One finds that at fixed ρ0,pulse widths tend to decrease with increasing ρsat.

One of the most notable aspects of the preceding analy-sis is that a closed-form prediction for the soliton profile

FIG. 1: (color online) Bright soliton intensity distributionsobtained by solving Eq. (9a) for ρ0 = 1.0. These profiles areuniversal since they are insensitive to variations in κ, α, andV0b. The dilation (pulse broadening) effect appears for

anomalous GVD (s = +1) when considering these curves asfunctions of τ rather than of ξ [12].

can be obtained. It is surprising, then, that while thecomplicated nonlinearity of Wood et al. [28] yields ex-act results, much simpler variants do not [29]. Indeed,Gatz and Herrmann have shown that NLS-type brightsolitons of two other saturable refractive-index models(involving two-level atoms [25] and double-doping [27])may be obtained only in the form of integral equations(to be solved iteratively) or through direct numerical so-lution of the quadrature problem. Similar intractabilityis also present in spatiotemporal regimes since the sys-tem of ordinary differential equations often turns out tobe the same (e.g., see Appendix B) but where ξ and thelocal-time coordinate are effectively interchangeable.

B. Intrinsic velocity

One must now ensure that Eq. (6b) is also rigorouslysatisfied where, for Ψb = 0, it assumes the simple form

(α− 2κKbV0b)dρb

dξ= 0. (10a)

In order for Eq. (10a) to hold true for arbitrary gradientsdρb/dξ, one must have α− 2κKbV0b = 0, or equivalentlyV0b = α/2κKb. Hence,

V0b = ± α√1 + 4κβb

(10b)

and where the structure of V0b here is identical to that forother known bright spatiotemporal solitons [11, 12]. Sucha structure arises from the space-time symmetry inherentto the linear part of the wave operator in Eq. (1). Incontrast, the connection between βb and ρ0 [c.f., Eq. (7c)]depends upon the details of the system nonlinearity.

Since the pulse travels (in dimensional units) at a ve-locity that is proportional to 1/V0b (a quantity defined indimensionless units), one can immediately see that spa-tiotemporal solitons must be assigned speeds that have

5

a weak dependence on intensity. This type of effect isnot typically encountered in NLS-type models, but itis nonetheless present in other universal nonlinear waveequations. For instance, the canonical Korteweg de-Vriesmodel with third-order linear dispersion and a quadraticself-steepening term predicts a fundamental sech2-shapedsoliton with an amplitude that is directly proportionalto its speed (an essential component for describing wave-breaking phenomena) [1].

IV. DARK SOLITON PULSES

One expects dark solitons to exist in the case of nor-mal GVD (where s = −1). Since the localized compo-nent of u resides as a dip travelling across a backgroundcontinuous-wave (cw) solution, we set ρ(ξ) → ρd(ξ),Ψ(ξ) → Ψd(ξ), and V0 → V0d where the ‘d’ subscriptdenotes dark solitons. The remaining two parametersare labelled as β → βcw and K → Kcw with reference tothe cw component.

A. Continuous-wave solutions

The cw solutions of Eq. (1) have a uniform intensityρ0 and may be assigned a finite frequency shift Ω so that

ucw(τ, ζ) = ρ1/20 exp [i(−Ωτ +Kcwζ)] exp

(−i ζ

2κ

)(11a)

and |ucw|2 ≡ ρ0. Substitution of ansatz (11a) in Eq. (1)yields the dispersion relation

κK2cw − Ω

(α− sΩ

2

)− 1

4κ= βcw, (11b)

which is parametrized by

βcw ≡2 + ρ0/ρsat

(1 + ρ0/ρsat)2

(ρ0

2

). (11c)

Note that dispersion relation (11b) is either elliptic (whens = +1) or hyperbolic (when s = −1) and thus alwayshas two branches corresponding to propagation in theforward and backward longitudinal senses [11].

The stability of the cw background, which is essentialfor the dark soliton, can be addressed through linear anal-ysis (see Appendix C for details with regard to genericnonlinearity functionals). Solution (11a) is perturbedby a small-amplitude periodic spatiotemporal modula-tion with temporal frequency Ωp. Any long-wavelengthdisturbance then grows whenever

Ω2p

2− 2s

ρ0

(1 + ρ0/ρsat)3 < 0. (12)

Instability is clearly present in the anomalous-GVDregime, but it disappears entirely for normal GVD since

inequality (12) can never be satisfied for any ρ0 ≥ 0 (de-tails are given in Appendix C). Hence, the cw back-ground of spatiotemporal dark solitons in saturable sys-tems have the desired modulational stability properties.

B. Quadrature equations

Having established the MI characteristics of the cwsolutions to Eq. (1), we now turn our attention to itsdark solitons. The quadrature equations are

d

dρd

[1

ρd

(dρd

dξ

)2]

= 4

(dΨd

dξ

)2

− 8

(α− 2κKcwV0√

1− 2κV 20d

)dΨd

dξ

− 8

[βcw −

2 + ρd/ρsat

(1 + ρd/ρsat)2

(ρd

2

)](13a)

and

d

dξ

[(dΨd

dξ− α− 2κKcwV0d√

1− 2κV 20

)ρd

]= 0. (13b)

Integration of Eq. (13b) leads to a result for the phasederivative,

dΨd

dξ=

(α− 2κKcwV0d√

1− 2κV 20d

)+c1d

ρd, (14a)

where c1d is a constant to be determined by consideringthe behaviour of Ψd as ξ → ±∞. Eliminating dΨd/dξterms from Eq. (13a) using Eq. (14a) leaves an equationsolely for the intensity quadrature:

d

dρd

[1

ρd

(dρd

dξ

)2]

= 4c21d

ρ2d

− 4

(α− 2κKcwV0d√

1− 2κV 20d

)2

− 8

[βcw −

2 + ρd/ρsat

(1 + ρd/ρsat)2

(ρd

2

)].

(14b)

Equations (14a) and (14b) are the dark-soliton analoguesof Eqs. (7a) and (10a) describing bright solitons. Theycan be solved to provide an exact solution.

6

C. Intensity quadrature

Integration of Eq. (14b) with respect to ρd gives(dρd

dξ

)2

= 4ρ2sat

(ρ2

d

ρsat+

ρd

1 + ρd/ρsat

)

− 4

2βcw +

(α− 2κKcwV0d√

1− 2κV 20d

)2 ρ2

d

+ c2dρd − 4c21d. (15)

By writing (dρd/dξ)2 = 4D(ρd − ρ1)(ρ0 − ρd)2/(1 +

ρd/ρsat), one may expand the numerator and comparethe coefficients of ρnd with n = 3, 2, 1, and 0. The resultis a system of four auxiliary algebraic equations:

D ≡ 1 +c3d

ρsat, (16a)

−D (2ρ0 + ρ1) ≡ c2d

4ρsat+ ρsat + c3d, (16b)

D(ρ2

0 + 2ρ0ρ1

)≡ c2d

4− c21d

ρsat+ ρ2

sat, (16c)

Dρ20ρ1 ≡ c21d, (16d)

and where we have introduced the parametrization

c3d ≡ −

2βcw +

(α− 2κKcwV0d√

1− 2κV 20d

)2 (16e)

for compactness. Equations (16a)−(16d) can be solvedsequentially to yield algebraic expressions for constantsD, c2d and c1d:

D =

(1 +

ρ0

ρsat

)−2(1 +

ρ1

ρsat

)−1

, (17a)

c2d = 4D

[ρ2

0

(1 +

ρ1

ρsat

)+ 2ρ0ρ1

]− 4ρ2

sat, (17b)

c21d =ρ2

0ρ1

(1 + ρ1/ρsat) (1 + ρ0/ρsat)2 , (17c)

though only D and c1d are used from here onwards. Itthen follows that Eq. (15) may be replaced by the factor-ized form(

dρd

dξ

)2

= 4(ρ0 − ρd)2(ρd − ρ1)

(1 + ρ0/ρsat)2 (1 + ρ1/ρsat) (1 + ρd/ρsat).

(18)In the domain where ξ > 0, the intensity gradient dρd/dξmust be positive. With these signs in mind, one canperform an integration of Eq. (18) to arrive at an implicit

spatiotemporal solution for ρd(ξ) that we express in theform first presented by Krolikowski and Luther-Davies[30]:(

ρ0 + ρsat

ρ0 − ρ1

)1/2

tanh−1

[(ρ0 + ρsat

ρ0 − ρ1

)1/2

ηd

]

− tanh−1 (ηd) =ρ

1/2sat

(1 + ρ0/ρsat) (1 + ρ1/ρsat)1/2

ξ,

(19a)

where

ηd[ρd] ≡(ρd − ρ1

ρd + ρsat

)1/2

(19b)

is a positive real parameter and ηd(ξ) ≡ ηd[ρd(ξ)]. Equa-tions (19a) and (19b) describe the exact (although im-plicit) dark soliton intensity profile, and they may besolved numerically to yield ρd(ξ) (see Fig. 2). For fixedρ0 and ρ1, the width of the pulse increases as ρsat de-creases. The physical nature of this inverse relationshipfollows directly from the nonlinearity-dispersion balancerequired for stationary states to exist (any reduction inself-phase modulation must be accompanied by a com-mensurate weakening of linear spreading) [1, 16].

D. Intrinsic velocity

An expression for the intrinsic velocity V0d can be de-rived by respecting the asymptotic behaviour of the so-lution. As ξ → ±∞, the intensity tends towards its cwvalue ρ0 and the phase gradient dΨd/dξ → 0 tends tozero. From Eq. (14a), it thus follows that

α− 2κKcwV0d√1− 2κV 2

0d

= −c1d

ρ0. (20a)

Agreement between Eqs. (16a) and (16b) demands thatρsat(1−B)−2βcw = Bρ1 (a result that can be confirmedalgebraically). It is then straightforward to show thatV0d satisfies the quadratic equation[

(2κKcw)2 + 2κV 20d loc

]V 2

0d

− 2α(2κKcw)V0d + (α2 − V 20d loc) = 0, (20b)

where we have defined V0d loc ≡ c1d/ρ0. Analysis of thetwo roots of Eq. (20b) must be performed with the twobranches of Kcw in mind [c.f., Eq. (11b)]. Under theseconditions, and being careful to choose the correct signfor forward- (+) and backward-travelling (−) solitons,one finds that the intrinsic velocity is given by

7

V0d(F ) = ±V0d loc(F )

1 + 2κρ0 (1 + ρ0/ρsat)

−2[2 + ρ0/ρsat + F 2

(1 + F 2ρ0/ρsat

)−1]− 2κα2

1/2

+ α√

1 + 4κβcw

1 + 2κρ0 (1 + ρ0/ρsat)−2[2 + ρ0/ρsat + F 2 (1 + F 2ρ0/ρsat)

−1] ,

(21a)

where the physical interpretation of

V0d loc(F ) ≡ ρ1/20 F

(1 + ρ0/ρsat) (1 + F 2ρ0/ρsat)1/2

(21b)

will become clear in Sec. VI B. Here, the dark solitoncontrast parameter F 2 ≡ ρ1/ρ0, with A2 + F 2 = 1 and0 < F 2 ≤ 1, has been introduced for universal notation[5, 11]. Inspection of Eq. (21b) also shows that, unlikefor the competing cubic-quintic nonlinearity [12], there isgenerally no upper limit on the allowed value of ρ0.

E. Phase quadrature

By combining Eqs. (14a) and (20a), it can be shownthat the soliton phase Φd(ξ) is given by the integral

Ψd(ξ) =

(c1d

ρ0

)∫dξ

(ρ0 − ρd

ρd

)+ Ψd0, (22a)

where Ψd0 is a constant that can be set to zero withoutloss of generality. The intensity profile is not known ex-plicitly as a function of ξ, but by deploying Eq. (18) anexact expression for the phase can be obtained whereinthe integration is over ρd. In the domain ξ ≥ 0,

Ψd [ηd] = tan−1

[(1

F

)(ρsat

ρ0

)1/2

ηd

]

+ F

(ρ0

ρsat

)1/2

tanh−1 (ηd) (22b)

and we note that Ψd(−ξ) = −Ψd(ξ). As expected, theparity of the dark soliton is determined by the sign of F ,

so under the change F → −F the phase profile is reversedin the ξ coordinate. The phase change across the darksoliton, ∆Ψd ≡ Ψd(+∞)−Ψd(−∞), can be expressed as

∆Ψd = π − 2 tan−1

[(F

A

)(1 +

ρ0

ρsat

)1/2]

+ 2F

(ρ0

ρsat

)1/2

× tanh−1

[A

(ρ0

ρsat

)1/2(1 +

ρ0

ρsat

)−1/2].

(23)

Under some conditions, ∆Ψd can be greater than π radi-ans (regimes in which the solution has been designated“darker than black” by Krolikowski et al. [33]).

V. MORE GENERAL SOLUTIONS

A. Frequency-velocity relations

The covariance of Eq. (1) under rotations or skews ofthe space-time coordinate axes means that one may al-ways observe the waves it describes from any arbitrarily-defined frame of reference. This degree of freedom per-mits one to construct more general solution families thatinvolve a finite frequency shift, denoted here by Ω, sothat now ub ∝ exp(iΩτ) and ud ∝ exp(−iΩτ).

Following the methods established in Refs. [11] and[12], one may relate Ω to a transformation parameter Vb

or Vd characterizing the coordinate change according to

Vb,d(Ω) =(Ω + α)

√1 + 4κβb,cw − 4sκΩ (α+ Ω/2)− α

√1 + 4κβb,cw

1 + 4κβb,cw − 2sκ (Ω + α)2 , (24)

and where Vb,d has a status analogous to the trans-verse velocity typically introduced into the analysis ofobliquely-propagating nonlinear beams [7].

In the spatiotemporal formulation, velocities can beshown to combine geometrically (akin to those in specialrelativity [13], as determined by the Lorentz rule) ratherthan additively (as in Galilean relativity). Hence, thenet velocity parameters Wb,d defined in the laboratoryframe, being the resultant of the intrinsic and transverse

contributions, can be obtained through a rule that is ei-ther pseudorelativistic (when s = +1 and the geometryis thus Euclidean) or relativistic (when s = −1 and thegeometry is instead Riemannian) in nature [6]:

Wb,d =V0b,0d + Vb,d

1− 2sκV0b,0dVb,d, (25a)

After some algebra, combining Eqs. (10b), (24) and (25a)one can show that the net velocity of bright solitons has

8

the compact form

Wb = ± α+ Ω√1 + 4κβb − 4κΩ

(α+ 1

2Ω) , (25b)

where the ± sign denoted forward- (+) and backward-travelling waves (−). An analogous expression for Wd issomewhat cumbersome and does not exhibit such alge-braic simplification.

Having introduced a transverse velocity, the lumpedspace-time variable ξ describing the intensity and phasequadratures [c.f. Eqs. (9a) and (19a)] must be replacedwith its transformed counterpart,

Θb,d(τ, ζ) ≡ τ ∓Wb,dζ√1 + 2sκW 2

b,d

, (26)

where Θb is associated with s = +1 and bright solution,while Θd is selected when considering s = −1 and darksolutions. For notational convenience, the ± sign inher-ent to Wb,d [e.g., see Eq. (25b)] has been absorbed intothe Θb,d variable so as the net velocity parameter is now

always a positive quantity. The frequency-shifted brightsoliton of Eq. (1) is now given by

ub(τ, ζ) = ρ1/2b (τ, ζ)

× exp

[iΩτ ± i

√1 + 4κβb − 4κΩ

(α+

Ω

2

)ζ

2κ

]

× exp

(−i ζ

2κ

), (27a)

where ρb(τ, ζ) is obtained from

tan−1 [ηb(τ, ζ)]

+1

2

(ρsat

ρ0

)1/2

ln

[(ρ0/ρsat)

1/2 + ηb(τ, ζ)

(ρ0/ρsat)1/2 − ηb(τ, ζ)

]=

ρ1/2sat

(1 + ρ0/ρsat)1/2

Θb(τ, ζ). (27b)

and ηb retains its formal definition from Eq. (9b). Simi-larly, the frequency-shifted dark soliton is

ud(τ, ζ) = ρ1/2d (τ, ζ) exp

[i

tan−1

[(1

F

)(ρsat

ρ0

)1/2

ηd(τ, ζ)

]+ F

(ρ0

ρsat

)1/2

tanh−1 [ηd(τ, ζ)]

]

× exp

[−iΩτ ± i

√1 + 4κβcw + 4κΩ

(α+

Ω

2

)ζ

2κ

]exp

(−i ζ

2κ

), (28a)

where (ρsat

ρ0

)1/2(1 + ρ0/ρsat)

1/2

Atanh−1

[(ρsat

ρ0

)1/2(1 + ρ0/ρsat)

1/2

Aηd(τ, ζ)

]

− tanh−1 [ηd(τ, ζ)] =ρ

1/2sat

(1 + ρ0/ρsat) [1 + (1−A2)ρ0/ρsat]1/2

Θd(τ, ζ) (28b)

and ηd still has the formal definition from Eq. (19b).Both solutions have now been parametrized by the ratioρ0/ρsat, which faciliates a more straightforward asymp-totic analysis of waveforms in the weak-saturation limit[defined to be ρ0/ρsat O(1)].

B. Non-degenerate bistability

As with cubic-quintic systems [12, 34, 35], it is possibleto sweep across the solution continua for the saturablesystem and identify pairs of non-degenerate bistablepulses. For the case of bright solitons, there emerges arange of values for ρsat within which two solitary pulsesmay have different peak intensities while sharing the

same full-width-half maximum (FWHM) [29, 30]. Sucha bistable characteristic is different from that proposedby Kaplan [36] describing degenerate solitons, where theintegrated intensity may become a multi-valued functionof the propagation constant. It is also distinct from thefamiliar S-shaped response curve from external feedbackin driven nonlinear cavities [14] (e.g., as described by theroots of a cubic equation modelling the steady states of aplane wave experiencing interferomic mistuning, periodicpumping, and coupling-mirror losses).

Non-degenerate bistable bright solitons with a FWHMof 2ν∆ (in the rest frame of the pulse), where ∆ =sech−1(2−1/2) ' 0.8814 is scale factor, are described byρb(Θb = ν∆) = ρ0/2 [34]. Applying that condition tosolution (27b) shows that ρ0 and ρsat can be connected

9

FIG. 2: (color online) Dark soliton intensity distributionsobtained by solving Eq. (19a) with ρ0 = 1.0 and ρ1 = 0.3.These profiles are universal since they are insensitive to κ,α, and V0b. The contraction (pulse narrowing) appears fornormal GVD (s = −1) when considering these curves as

functions of τ rather than of ξ [12].

by the implicit equation [29]

tan−1

[(ρ0/ρsat

2 + ρ0/ρsat

)1/2]

+1

2

(ρsat

ρ0

)1/2

ln

[(2 + ρ0/ρsat)

1/2+ 1

(2 + ρ0/ρsat)1/2 − 1

]

=

(ρsat

1 + ρ0/ρsat

)1/2

ν∆. (29a)

For weak saturation effects, the lower branch tends to-wards ρ0 ' 1/ν2 while for strong saturation [character-ized by ρ0/ρsat O(1)], the upper branch diverges ac-cording to ρ0 ' ρsat

[ρsat(4ν∆/π)2 − 1

](see Fig. 3).

For dark solitons, the bistability condition changesslightly to read ρd(Θd = ν∆) = (ρ0 +ρ1)/2 [35], in whichcase application to solution (28b) leads to a second im-plicit equation [30]

(ρsat

ρ0

)1/2(1 + ρ0/ρsat)

1/2

A

× tanh−1

[(ρsat

ρ0

)1/2(1 + ρ0/ρsat

2−A2 + 2ρsat/ρ0

)1/2]

− tanh−1

[A

(2−A2 + 2ρsat/ρ0)1/2

]

=ρ

1/2sat

(1 + ρ0/ρsat) [1 + (1−A2)ρ0/ρsat]1/2

ν∆.

(29b)

Equation (29b) prescribes pairs of gray solitons that havethe same FWHM but where the cw backgrounds havedifferent intensities. The lower branch tends towardsρ0 ' 1/ν2A2 in the weak saturation regime (see Fig. 4),as expected [30].

VI. ASYMPTOTIC ANALYSES

A. Soliton hierarchies

For waves of low intensity, defined by |u|2/ρsat O(1),saturation is relatively weak and the system response istraditionally represented through a truncated power se-ries [18]. To leading order, the dominant nonlinear con-tribution to Eq. (1) is well approximated by

1

2

2 + |u|2/ρsat

(1 + |u|2/ρsat)2 |u|

2u '(

1− 3

2ρsat|u|2)|u|2u, (30)

which corresponds to a cubic-quintic model with a domi-nant (positive) cubic term and a small (negative) quinticcorrection. In the notation of Ref. [12], one can iden-tify coefficients γ2 ≡ +1 and γ4 ≡ −3/2ρsat with thestandard cubic nonlinearity clearly recovered when thequintic contribution is neglected. Hence, we expect spa-tiotemporal saturable solitons to transition through theircubic-quintic countparts when intensities are much lessthan ρsat and the cubic limit is approached. It is self-evident that the underlying quadrature equations mustalso reduce accordingly at each stage.

1. Bright solitons

To facilitate the asymptotic analysis of bright solitons,it is helpful to recast exact solution (27b) in the slightlydifferent but more instructive form [37]

2

(ρ0

ρsat

)1/2

tan−1

[(ρ0

ρsat

)1/2(1− ρb/ρ0

1 + ρb/ρsat

)1/2]

+ cosh−1

[2ρ0 (1− ρ0/ρsat)

−1 − ρb

ρb (1 + ρ0/ρsat) (1− ρ0/ρsat)−1

]= 2√

2βbΘb(τ, ζ), (31a)

which more closely resembles the target cubic-quinticbright soliton [12]. For ρ0/ρsat O(1), result (31a) iswell approximated by

2

(ρ0

ρsat

)(1− ρb

ρ0

)1/2

+ cosh−1

[2ρ0 (1 + ρ0/ρsat)− ρb

ρb (1 + 2ρ0/ρsat)

]' 2√

2βbΘb(τ, ζ).

(31b)

In going from Eq. (31a) to (31b), one is obliged to rec-ognize the leading-order change to the βb parameter [c.f.Eq. (7c)], namely βb ' (1 − ρ0/ρsat)(ρ0/2). Hence, thenet velocity Wb [defined in Eq. (25b)] has the correctbehaviour so that the right-hand side of Eq. (31b) nat-urally converges. Note that ρ0/ρsat does not need to beall that small before the left-hand side of Eq. (31b) can

10

be replaced to yield:

cosh−1

[2ρ0 (1− ρ0/ρsat)− ρb

ρb (1− 2ρ0/ρsat)

]' 2√

2βbΘb(τ, ζ)

(31c)[for instance, approximate solutions (31b) and (31c) arenearly indistinguishable for ρ0/ρsat = 1/10]. Crucially, itfollows that the approximate saturable soliton describedby Eq. (31c) corresponds to an exact cubic-quintic solitonwith γ2 = +1 and γ4 ≡ −3/2ρsat [12]. As ρ0/ρsat → 0,the cubic result also emerges from Eq. (31c) [11],

cosh−1

(2ρ0 − ρb

ρb

)' 2ρ

1/20 Θb(τ, ζ), (31d)

as it must. Hence we have proved that bright solitonsform a hierarchy wherein saturable solutions must neces-sarily pass through the corresponding cubic-quintic shapebefore finally (and rather slowly) converging on the cubiclimit as ρ0/ρsat → 0.

2. Dark solitons

A similar analysis of dark solitons is more involved, butthe same procedure is generally followed for its intensityand phase quadratures. Exact solution (28b) describingthe intensity distribution can be reexpressed [37] as

cosh−1

2ρ0A

2 (1 + ρ0/ρsat)−[1 +

(1 +A2

)ρ0/ρsat

](ρ0 − ρd)

[1 + (1−A2) ρ0/ρsat] (ρ0 − ρd)

− 2A

(1 + ρ0/ρsat)1/2

(ρ0

ρsat

)1/2

tanh−1

1

ρ1/2sat

[ρd −

(1−A2

)ρ0

1 + ρd/ρsat

]1/2 = 2

√2βdΘd(τ, ζ), (32a)

and where we have introduced the parameter

βd ≡ρ0A

2

2

(1 +

ρ0

ρsat

)−3 [1 +

(1−A2

) ρ0

ρsat

]−1

(32b)

for a more transparent recovery of known results for thecubic-quintic system [12].

In the limit ρ0/ρsat O(1), the expression for βd re-

duces to its cubic-quintic counterpart with γ2 = +1 andγ4 = −3/2ρsat, namely βd = (ρ0A

2/2)[1−(4−A2)ρ0/ρsat][12]. Since the intrinsic velocity V0d derived in Ref. [12] isalso recovered from Eq. (21a) in the same way, it followsthat right-hand side of Eq. (32a) necessarily convergesas desired. As in the case of bright solitons, the ratioρ0/ρsat does not need to be especially small before theexact left-hand side of Eq. (32a) can be replaced with anexcellent approximation thus:

cosh−1

2ρ0A

2[1− (ρ0/ρsat)

(4−A2

)]− (1− 4ρ0/ρsat) (ρ0 − ρd)

[1− 2 (ρ0/ρsat) (2−A2)] (ρ0 − ρd)

' 2√

2βdΘd(τ, ζ). (33a)

One subsequently recovers the cubic-quintic intensityprofile detailed in Ref. [12]. As ρ0/ρsat → 0, the well-known profile for the cubic system [7] appears as thelimit of Eq. (33a):

cosh−1

[2ρ0A

2 − (ρ0 − ρd)

ρ0 − ρd

]' 2ρ

1/20 AΘd(τ, ζ). (33b)

The phase distribution of the cubic-quintic dark solitoncan be recovered from the saturable solution in a similarway (we do not present the details here). Hence, darksolitons must also form a self-consistent saturable—cubic-quintic—cubic hierarchy.

B. Slowly-varying envelopes

1. Envelope equation

By neglecting the first term in Eq. (1), we can recoverthe governing equation of conventional pulse theory inthe laboratory frame:

i

(∂u

∂ζ+ α

∂u

∂τ

)+s

2

∂2u

∂τ2+

1

2

2 + |u|2/ρsat

(1 + |u|2/ρsat)2|u|2u ' 0.

(34a)By Galilean boosting to the local time-frame, defined bycoordinates τloc ≡ τ − αζ and ζloc = ζ, Eq. (34a) istransformed into the more familiar form considered by

11

FIG. 3: (color online) Non-degenerate bistablity curves forsaturable bright solitons as predicted by Eq. (29a). Lower

branches with 1/ρsat → 0 correspond to regimes wheresaturation is weak, in which case the peak intensity

converges toward ρ0 ' 1/ν2.

Krolikowski and Luther-Davies [29, 30],

i∂u

∂ζloc+s

2

∂2u

∂τ2loc

+1

2

2 + |u|2/ρsat

(1 + |u|2/ρsat)2|u|2u ' 0. (34b)

Crucially, one must be able to recover the soliton so-lutions of these two related models in the limit that allcontributions from κ∂2u/∂ζ2 are sufficiently small simul-taneously. In practice, one is obliged to asymptote thespatiotemporal predictions for velocities and propagationconstants in conjunction with a Galilean transformationto the (τloc, ζloc) frame.

2. Intrinsic, transverse, and net velocities

We first consider algebraic results for the various ve-locity contributions (10b), (21b), and (25a) under the as-sumption of slowly-varying envelopes. The label ‘SVEA’is used to denote these components in the laboratoryframe, while ‘loc’ refers to their local-time frame rep-resentations.

In the limit κβb O(1), bright solitons have an in-trinsic velocity V0b ' α ≡ V0b SVEA. For dark solitonswith κβcw O(1) and κα2 O(1), it follows thatV0d ' V0d loc + α ≡ V0d SVEA. When considering the pa-rameters Vb,d, the additional inequality κ|Ω(α+Ω/2)| O(1) leads to the simple result Vb,d ' Ω ≡ VSVEA.This key finding demonstrates that transverse veloci-ties and frequency shifts are interchangeable in conven-tional pulse theory. Finally, the net velocities Wb,d arewell-approximated by Wb,d ' V0b,0d SVEA + VSVEA ≡Wb,d SVEA.

In the local time-frame, parameters V0b,0d SVEA andVSVEA combine in a way that maps directly onto the ve-locity combination rule of Galilean relativity. For brightsolitons, the intrinsic velocity V0b SVEA is transformedaway so that in (τloc, ζloc) coordinates one has V0b loc = 0.It then follows that Wb loc = VSVEA = Ω, and hence solu-tions with Ω = 0 describe pulses that are strictly station-ary (that is, they travel with their peak always centered

FIG. 4: (color online) Non-degenerate bistablity curves forsaturable dark solitons [(a) black (A = 1) and (b) gray (withν = 1) solutions] as predicted by Eq. (29b). Lower branches

with 1/ρsat → 0 correspond to weak saturation, in whichcase the cw intensity tends to ρ0 ' 1/ν2A2.

on τloc = 0). The situation is slightly more complicatedfor dark solitons. The factor α in V0d SVEA disappears sothat the intrinsic velocity of the dark soliton is simplyV0d loc and hence Wd loc = V0d loc + Ω. Black solutions(having F = 0 = V0d loc) with Ω = 0 thus have zero localnet velocity and are also strictly stationary.

3. Asymptotic solutions

Since all the coordinate transformations we have con-sidered here are geometrical operations, the profile of anysolution must be independent of the frame of reference inwhich it is observed. For example, a sech-shaped pulsein the laboratory frame must also be sech-shaped in thelocal-time frame (though in spatiotemporal regimes onenecessarily encounters a contraction or dilation factor)[6, 7].

From the results of the previous subsection, it fol-lows that under the SVEA, one must have Θb,d(τ, ζ) 'τ ∓ Wb,dζ ≡ Θb,d SVEA(τ, ζ). The intensity and phasequadratures of localized components of the solutions [c.f.,Eqs. (27b), (28a) and (28b)] remain formally unchangedwhen assuming slowly-varying envelopes (one simply re-places Θb,d with Θb,d SVEA). However, the linear part ofthe phase distribution alters its structure due to the co-ordinate change. When considering forward-propagatingspatiotemporal solitons, transforming to the local-time

12

frame shows that

ub(τloc, ζloc) ∝ exp

[iΩτloc + i

(βb −

Ω2

2

)ζloc

], (35a)

and

ud(τloc, ζloc) ∝ exp

[−iΩτloc + i

(βcw +

Ω2

2

)ζloc

].

(35b)

These solutions satisfy Eq. (34b) exactly for s = +1 ands = −1, respectively. Hence, one can now fully appreciatethat the known (conventional) solitons derived by Kro-likowski and Luther-Davies [29, 30] are important subsetsof the more general spatiotemporal solutions. Applyingthe same multiple-limit procedure to the backward spa-tiotemporal solitons yields largely similar results exceptthat a rapidly-varying phase term, exp[−i2(ζ/2κ)], sur-vives in the linear phase distributions. The non-vanishingnature of this contribution demonstrates that conven-tional pulse theory (based on a nonlinear-Schrodingerformalism) has no analogue of backward waves.

VII. SOLITON STABILITY

In going from cubic [7] to cubic-quintic [12] to sat-urable systems, the linear part of the wave operator al-ways has the same form. In that sense, one does notexpect to encounter substantial changes in soliton sta-bility characteristics since the fundamental elliptic / hy-perbolic structure of the governing equation remains un-changed. Here, the same physical arguments with re-gards frame-of-reference symmetries are deployed, andstandard tools [viz., the Vakhitov-Kolokolov (VK) andrenormalized-momentum integral criteria] that have pre-viously proved so invaluable [7, 12] are applied to makepredictions about the robustness of localized solutions toEq. (1).

A. Vakhitov-Kolokolov criterion

A bright-type solution of Eq. (34b) is predicted to bestable against small disturbances if the integrated pulseintensity P , defined by

P ≡+∞∫−∞

dτloc |ub|2, (36a)

has a positive gradient such that

d

dβbP (βb) > 0, (36b)

where βb is the propagation constant [c.f. Eq. (7c)]. Sincethe wave intensity profile is known only implicitly, the ex-plicit computation of P (βb) can be potentially awkward.

FIG. 5: (color online) Integrated intensity curves P (βb)obtained from Eq. (36c) for increasing saturation parameter.

The gradient dP/dβb is always positive so that the VKcriterion [c.f. Eqs. (36a) and (36b)] is satisfied.

However, the calculation can be facilitated indirectly bytransforming the integral [36] so that

P (βb) =1√2

ρ0(βb)∫0

dR0

[βb −

1

1 +R0/ρsat

(R0

2

)]−1/2

,

(36c)where ρ0(βb) = 2βb/(1 − 2βb/ρsat). Since the peak in-tensity lies in the range 0 ≤ ρ0 < ∞, it must be that0 ≤ βb < ρsat/2 (note that the corresponding cubic-quintic soliton also has a maximum allowed value for βb

[12]). Plotting the P (βb) curves reveals that inequality(36b) tends to be satisfied and hence saturable solitonsare anticipated to be stable entities (see Fig. 5).

B. Perturbed bright solitons

To test analytical predictions of bright soliton stability,a similar prescription is followed to that in Ref. [12]. Ini-tial data is selected for Eq. (1) using solution (27a) butwhere the dilation factor (1 + 2κW 2

b )1/2 characterizingthe broadening of ρb(τ, 0) in the anomalous-GVD regimeis omitted from Θb(τ, 0). The strength of the local shapeperturbation thus increases with Ω = 4, 8, 12 and 16.

Typical examples of pulse self-reshaping due to internaldynamics are shown in Fig. 6. A saturation intensity ofρsat = 4.0 has, according to Eq. (29a), bistable lower- andupper-branch solitons with peak intensities ρ0 ' 2.298and ρ0 ' 8.763, respectively. The lower-branch solutionexhibits monotonically-decaying oscillations as the wave-form evolves towards a stationary state of Eq. (1), with asmall amount of energy shed as low-amplitude dispersivewaves (radiation). The upper-branch solution behavessomewhat differently, with persistent (and much morerapid) oscillations surviving in the long-term evolution.These oscillations are bounded within an envelope thatis weakly modulated in ζ—behaviour that is consistentwith the excitation of an internal mode [38]—with sim-ulations giving no indication of convergence towards a

13

stationary state (even over propagation distances muchlonger than those shown).

The self-reshaping characteristics of the lower-branchsoliton are reminiscent of those recently uncovered incubic-quintic systems [12]. By interpreting perturbation-induced radiative losses as a mechanism for local energydissipation (while noting that the system is still globallyconservative), one might regard the asymptotic station-ary states emerging in Fig. 6(a) as fixed-point attractorssurrounded by wide basins of attraction. Similarly, thesurviving oscillatory solutions in Fig. 6(b) are qualita-tively similar to the limit-cycle attractors reported else-where in spatial soliton contexts [7].

C. Renormalized-momentum criterion

The stability of dark solitons of generalized NLS-typemodels such as Eq. (34b) is typically discussed in termsof a renormalized momentum Mren,

Mren ≡i

2

+∞∫−∞

dτloc

(ud

∂u∗d∂τloc

− u∗d∂ud

∂τloc

)(1− ρ0

|ud|2

).

(37a)The integral expression for Mren can be recast in a moreconvenient form by way of Eq. (14a) and by noting that

FIG. 6: (color online) Evolution of the bistable brightsoliton peak amplitude when the initial waveform resides on

the (a) lower branch (ρ0 = 2.298) and (b) upper branch(ρ0 = 8.763)—c.f. Fig. 3 with ν = 1.0. System parameters:ρsat = 4.0, s = +1, α = 1.0, κ = 1.0× 10−3. Blue circle:

Ω = 4. Green square: Ω = 8. Red triangle: Ω = 12. Blackdiamond: Ω = 16.

ξ and τloc are interchangeable in conventional analyses:

Mren = −V0d loc

+∞∫−∞

dτloc(ρd − ρ0)

2

ρd. (37b)

Solitons are predicted to be stable when the inequality

d

dV0d locsMren(V0d loc) > 0 (37c)

is satisfied. Note that Eq. (37c), proved by Barashenkov[39] to be an acceptable stability criterion and furtherdeveloped by Pelinovsky et al. [40], captures nonlineardynamical phenomena that the MI (linear) calculation ofSec. IV A (and Appendix C) clearly cannot.

By considering the behaviour of the integral inEq. (37b), Kivshar and Afanasjev have shown that blacksolitons of Eq. (34b) possess a drift instability that ap-pears when ρsat/ρ0 is less than a critical value of ≈ 0.45(see Fig. 3 in Ref. [41]). The instability manifests itself asthe gradual transformation of an initially black (F = 0)solution into a gray (|F | > 0) waveform, accompanied bythe strong emission of radiation. In the following simu-lations with Eq. (1), we restrict our attention to param-eter regimes above criticality. A set of typical curves forthe renormalized momentum, as defined in Eq. (37b), isshown in Fig. 7. The gradients are generally positive andMren is undefined at V0d loc = 0 [41].

D. Perturbed dark solitons

The perturbed dark soliton initial-value problem is de-fined by using solution (28a) but where the contractionfactor (1−2κW 2

d )1/2 in the normal-GVD regime is omit-ted from Θd(τ, 0). Attention is first paid to black solitonswith frequency shifts of Ω = 4, 8, 12, and 16. The tempo-ral width of the input pulse, defined as w0 ≡ (2βd)−1/2

FIG. 7: (color online) Renormalized dark solitonmomentum when ρ0 = 2.0, computed with Eq. (37b) for the

contrast range 0 < F ≤ 1. The gradient dMren/dV0d loc isalways positive for V0d loc > 0, so that stability criterion

(37c) is always satisfied. Here, Mren approaches a numericalvalue of −2π as V0d loc tends toward zero.

14

FIG. 8: (color online) Evolution of the bistable blacksoliton full width when the initial waveform resides on the

(a) lower branch (ρ0 = 2.383) and (b) upper branch(ρ0 = 6.167)—c.f. Fig. 4 with ν = 1.0 (horizontal barsindicate theoretical predictions). System parameters:ρsat = 5.0, s = −1, α = 1.0, κ = 1.0× 10−3. Blue circle:

Ω = 4. Green square: Ω = 8. Red triangle: Ω = 12. Blackdiamond: Ω = 16.

[c.f. Eq. (32b)], is broader than that required by the exactsolution. We thus expect the localized pulse to becomenarrower as it travels through space, evolving smoothlytowards its limiting value of w∞ = w0(1 − 2κW 2

d )1/2.Typical self-reshaping characteristics are shown in Fig. 8for ρsat = 5.0 and ν = 1.0 [in which case Eq. (29b) deter-mines the lower- and upper-branch cw intensities to beρ0 ' 2.383 and ρ0 = 6.167, respectively]. The evolutioninto a stationary state occurs relatively rapidly, within adistance of ζ ' 10.

Finally, illustrative results are presented for perturbedgray solitons (curves for the reshaping pulse widths aresimilar to those in Fig. 8 but they occur on much longerscalelengths in ζ). Such solutions tend not to preservetheir grayness and are described by F → F (ζ), whereF (ζ) must be obtained numerically. Gray solitons thusoften exhibit a small drift-type instability that is reminis-cent of that in cubic-quintic systems [12], where the min-imum intensity of the pulse dip (which is itself varyingslowly in ζ) tends to travel along the (approximately lin-ear) characteristic τ−Wd(ζ)ζ = const. and where V0d(ζ)is computed from Eq. (21a) after adiabatic fitting of thenumerical dataset to solution (28b). Typical variationsin relaxing grayness for perturbed solitons with a fixedsaturation intensity ρsat = 5.0 are shown in Fig. 9. Sim-ulations have also shown that for a fixed perturbation

FIG. 9: (color online) Evolution of the gray solitoncontrast parameter when the initial waveform has ρ0 = 2.0and F 2(0) = 0.5. System parameters: ρsat = 5.0, s = −1,α = 1.0, κ = 1.0× 10−3. Blue circle: Ω = 4. Green square:

Ω = 8. Red triangle: Ω = 12. Black diamond: Ω = 16.

FIG. 10: (color online) Evolution of the gray solitoncontrast parameter when the initial waveform has ρ0 = 2.0and F 2(0) = 0.5 and the initial perturbation has Ω = 8.0.

System parameters: s = −1, α = 1.0, κ = 1.0× 10−3. Bluecircle: ρsat = 1.0. Green square: ρsat = 2.0. Red triangle:

ρsat = 4.0. Black diamond: ρsat = 8.0.

strength (e.g., for a moderate value Ω = 8.0), the relax-ation of F (ζ) tends to occur over distances that decreasewith increasing ρsat (see Fig. 10).

VIII. CONCLUSIONS

Our analyses of spatiotemporal systems have now con-sidered exact bright and dark envelope solitons in somedetail for the three classic nonlinearities where one mightexpect to find analytical solutions—cubic [11], cubic-quintic [12], and saturable models. These wavepacketstend to exhibit a raft of corrections to their known con-ventional counterparts that arise solely from the space-time-symmetric nature of the governing equation; theyinclude relativistic- or pseudorelativistic-type contrac-tion/dilation factors in the pulse width, along withgeneric (intensity- and frequency-dependent) modifica-tions to propagation constants and group velocities. Ineach case, standard mathematical tools have been usedto investigate spatiotemporal wave stability, with sup-porting simulations testing and verifying theoretical pre-

15

dictions by way of perturbed initial-value problems. Thenew solitons have demonstrated remarkable robustnessthroughout, and they may be interpreted as attractorsin the system’s dynamics.

Having derived soliton solutions (fundamental station-ary states to facilitate subsequent investigations), thereis now a wide spectrum of higher-order effects to explorewithin the spatiotemporal context. Quite general wave-based phenomena to accommodate include third- [42] andfourth-order [43] linear dispersion while, in the photonicsdomain specifically, self-steepening and stimulated Ra-man scattering [44, 45] are often practical concerns inoptical-fiber systems. The arena of spatiotemporal soli-ton collisions is also one that merits attention, given thefundamental importance of nonlinear wave interactionsin general [46, 47] and for optical applications in partic-ular [48].

Finally, it is desirable to extend our (1 + 1)D spa-tiotemporal modelling into higher-dimensional (2 + 1)Dand (2 + 2)D regimes, where excitations may be fullylocalized in four-dimensional space-time. Such an ex-ercise opens the door to formulating symmetrized (i.e.,more complete relativistic- and pseudorelativistic-type)descriptions of exotic phenomena such as optical bullets[8] and X waves [49] that include the interplay betweennonlinearity, GVD, and Helmholtz (as opposed to parax-ial) diffraction.

Appendix A: Example in waveguide optics

We consider a scalar electric field of the form E(t, z) =A(t, z) exp[i(k0z − ω0t)] + c.c., where A(t, z) is the en-velope and “c.c.” denotes the complex conjugate of thepreceding quantity. The travelling wave has a center fre-quency ω0 and propagation constant k0 = n0ω0/c, wheren0 is the linear refractive index of the host medium atω0 and c is the speed of light in vacuo. Taking the waveintensity to be |A(t, z)|2, the generic model chosen for asaturable nonlinear refractive index nNL is [28]

nNL(|A|2) =n2Isat

2

[1− 1

(1 + |A|2/Isat)2

], (A1)

where Isat is the saturation intensity parametrizing theresponse. At low intensities, where |A|2/Isat O(1),saturation is small and Eq. (A1) is well approximated bynNL(|A|2) ' n2|A|2 + n4|A|2, where n4 ≡ −(3/2)n2/Isat

(the system essentially has a cubic-quintic response) [18].Hence, the standard Kerr effect, nNL(|A|2) = n2|A|2 de-termines the dominant behaviour when |n4||A|2 is negli-gible compared with n2.

Deployment of the Fourier decomposition techniquesto capture the linear-dispersive properties of the system

[16] then leads to the following equation for A:

1

2k0

∂2A

∂z2+ i

(∂A

∂z+ k1

∂A

∂t

)− k2

2

∂2A

∂t2

+ω0

c

(n2

2

) 2 + |A|2/Isat

(1 + |A|2/Isat)2 |A|

2A = 0. (A2)

The two parameters k1 ≡ (∂k/∂ω)ω0= 1/vg and k2 ≡

(∂2k/∂ω2)ω0are related to the group-velocity and tem-

poral dispersion (GVD), respectively, where k is formallyobtained from an associated eigenvalue problem (i.e.,solving Maxwell’s equations for the transverse distribu-tion of the guided field [50]).

One can now introduce a scaling for the laboratoryspace z and time t coordinates according to ζ = z/L andτ = t/tp, and also for the envelope A(t, z) = A0u(t, z).By connecting the spatial and temporal units throughL ≡ t2p/|k2| (essentially scaling to a linearly-dispersingreference Gaussian pulse [50]) and measuring amplitudesin units defined by (ω0/c)n2A

20L ≡ 1, we can arrive

at Eq. (1) where the parameters are κ = 1/2k0L, α =k1L/tp, s = −sgn(k2), and ρsat = Isat/A

20.

Appendix B: Two-level atom saturablerefractive index model

1. Bright solitons

Perhaps the simplest saturation model stems from thetwo-level-atom approximation [14]. Here, it is capturedwithin the envelope equation

κ∂2u

∂ζ2+ i

(∂u

∂ζ+ α

∂u

∂τ

)+s

2

∂2u

∂τ2+

1

1 + |u|2/ρsat|u|2u = 0,

(B1)which is a direct spatiotemporal generalization of theclassic model considered by Gatz and Herrmann [25]. Forweak saturation, where |u|2/ρsat O(1), the nonlinear-ity functional tends to the constant ρsat [c.f. a limit-ing value of ρsat/2 for Eq. (1)]. Moreover, a competingcubic-quintic response emerges in which γ2 = +1 cap-tures the dominant contribution and γ4 ≡ −1/ρsat (c.f.γ4 ≡ −3/2ρsat in Sec. VI A) parametrizes the leading-order correction to the Kerr effect [12, 18].

Following the method detailed in Sec. II, the intensityquadrature is described by

d

dρb

[1

ρb

(dρb

dξ

)2]

= 8

(βb −

ρb

1 + ρb/ρsat

)(B2a)

so that, after a first integration, we find(dρb

dξ

)2

= 8βbρ2b − 8ρsatρ

2b

[1− ρsat

ρbln

(1 +

ρb

ρsat

)]+ c2bρb. (B2b)

16

Applying the bright-soliton boundary conditions at ξ →±∞ shows that c2b = 0 (recall that the correspondingconstant was non-vanishing, assuming a value of 4ρ2

sat,in Sec. III A). Considering the boundary conditions atξ = 0 also leads to

βb = ρsat

[1− ρsat

ρ0ln

(1 +

ρ0

ρsat

)]. (B3)

Eliminating βb from Eq. (B3) using Eq. (B2b) leads to(dρb

dξ

)2

= 8ρ2satρ

2b

×[

1

ρbln

(1 +

ρb

ρsat

)− 1

ρ0ln

(1 +

ρ0

ρsat

)],

(B4)

which cannot now be integrated exactly. To computethe intensity profile for a given peak intensity ρ0, one

might resort to direct numerical techniques [e.g., treatingEq. (B2a) as a boundary-value problem and applying theshooting method (see Fig. 11)]. Alternatively, Eq. (B4)can be transformed into an integral equation. In thedomain ξ > 0, where dρb/dξ < 0, it can be shown thatρb may be obtained from

ρb(ξ)

ρ0= exp

[− 2√

2ρsat

ξ∫0

dX

1

ρb(X)ln

[1 +

ρb(X)

ρsat

]

− 1

ρ0ln

(1 +

ρ0

ρsat

)1/2], (B5)

which may be solved iteratively (e.g., in tandem with thenon-degenerate bistability condition) [25].

After implementing the coordinate discussed in Sec. V,one can write down the more general frequency-shiftedbright soliton solution of Eq. (B1), namely

ub(τ, ζ) = ρ1/20 exp

[− sgn (Θb)

√2ρsat

Θb(τ,ζ)∫0

dX

1

ρ(X)ln

[1 +

ρ(X)

ρsat

]− 1

ρ0ln

(1 +

ρ0

ρsat

)1/2]

× exp

[iΩτ ± i

√1 + 4κβb − 4κΩ

(α+

Ω

2

)ζ

2κ

]exp

(−i ζ

2κ

), (B6)

where Θb(τ, ζ) and Wb are given by Eqs. (26) and (25b),respectively. We note in passing, but do not rigorouslyprove here, that the double-doping saturable nonlinear-ity proposed by Gatz and Herrmann [27] can be treatedin exactly the same way and their solution (expressed interms of an integral equation) generalized to spatiotem-poral regimes accordingly. Since bright soliton (B6) isnow known, it is now instructive to consider its stabilityproperties in relation to those of solution (27a).

2. Soliton stability

For completeness, we thus consider briefly the sameclass of perturbed bright soliton initial-value problem forEq. (B1) as in Sec. VII. The input pulse ub(τ, 0) isdefined by solution (B6) but where the dilation factor(1 + 2κW 2

b )1/2 is omitted from the upper limit [i.e., fromΘb(τ, 0)]. For straightforward comparison with earliersimulations, the same frequency shifts are retained tocontrol the strength of the perturbation (Ω = 4, 8, 12,and 16) and peak intensities ρ0 = 2.298 and ρ0 = 8.763are used (though these values do not necessarily lie onthe bistability curve [25]). We also note that for fixedpeak intensity and saturation parameters, the two-levelmodel tends to have a stronger nonlinear response than

the phenomenological model of Wood et al. [28].Comparing Figs. 12(a) and 6(a) shows that quantita-

tively similar reshaping curves can be expected for thelow-intensity waves. More pronounced differences can ap-pear for higher-intensity waves, where the long-term limitcycle-type oscillations associated with reshaping solitonsof Eq. (1) may be absent from the predictions of model

FIG. 11: (color online) Bright soliton intensitydistributions obtained by solving Eq. (B2a) numerically forρ0 = 1.0 (they are universal since they are insensitive to

variations in κ, α, and V0b). Differences between thestationary solutions of Eqs. (1) and (B1) occurpredominantly at low values of ρsat (c.f. Fig. 1).

17

FIG. 12: (color online) Evolution of the bright soliton peakamplitude when the initial waveform for Eq. (B1) has a peak

amplitude of (a) ρ0 = 2.298 and (b) ρ0 = 8.763. Systemparameters: ρsat = 4.0, s = +1, α = 1.0, κ = 1.0× 10−3.Blue circle: Ω = 4. Green square: Ω = 8. Red triangle:

Ω = 12. Black diamond: Ω = 16.

(B1) [compare Figs. 12(b) and 6(b)]. One finds, instead,oscillatory features (with variations on a shorter longi-tudinal scalelength) that vanish as ζ → ∞ to leave astationary solution. The simulations presented here andin Sec. VII B thus demonstrate that two different spa-tiotemporal saturable-nonlinearity models possess solitonsolutions with similar stability characteristics.

Appendix C: Modulational instability

1. Perturbation dispersion relation

Here, we analyze the MI characteristics for a fully-second-order envelope equation with a generic nonlinear-ity function f(|u|2),

κ∂2u

∂ζ2+i

(∂u

∂ζ+ α

∂u

∂τ

)+s

2

∂2u

∂τ2+ f(|u|2)u = 0, (C1a)

where f(0) = 0. Typical examples of f include cubic-quintic systems such as f(|u|2) = γ2|u|2 +γ4|u|4 [12], andsaturable systems that may have either a simple form,f(|u|2) = |u|2(1 + |u|2/ρsat)

−1 [c.f., Eq. (B1)], or onethat is more complicated,

f(|u|2) =1

2

2 + |u|2/ρsat

(1 + |u|2/ρsat)2 |u|

2 (C1b)

[c.f., Eq. (1)]. The two cw families are described by

ucw(τ, ζ) = ρ1/20 exp

(±i√

1 + 4κβcwζ

2κ

)exp

(−i ζ

2κ

),

(C2)where ρ0 ≡ |ucw|2 is the uniform intensity, βcw ≡ f(ρ0),and the ± sign determines the longitudinal sense of prop-agation (+ for forwards in ζ, − for backwards).

Perturbed solutions are now sought that have the form

u(τ, ζ) = ρ1/20 [1 + εa(τ, ζ)] exp

(±i√

1 + 4κβcwζ

2κ

)× exp

(−i ζ

2κ

), (C3a)

where ε O(1) is a small parameter and a(τ, ζ) is a com-plex function with O(1) magnitude describing perturba-tions to both the amplitude and phase of the underlyingcw solution. It thus follows that, within a linear approxi-mation, |u(τ, ζ)|2 ≡ ρ ' ρ0+ερ0 (a+ a∗) ≡ ρ0+∆ρ(τ, ζ).The nonlinearity function f can then be Taylor-expandedaround the cw intensity ρ0 so that

f(|u|2) ' f(ρ0 + ∆ρ) ' βcw + ερ0f′(ρ0) (a+ a∗) ,

(C3b)

where f ′(ρ0) ≡ df(ρ)/dρ|ρ=ρ0 parametrizes the leading-order correction [e.g., f ′(ρ0) = γ2 + 2γ4ρ0 for a cubic-quintic nonlinearity, or f ′(ρ0) = (1 + ρ0/ρsat)

−2 andf ′(ρ0) = (1 + ρ0/ρsat)

−3 for the two saturation models,respectively]. Higher-order terms in the expansion, suchas (∆ρ)2 and (∆ρ)3 etc., are neglected in linear analysis.

Substitution of Eqs. (C3a) and (C3b) into Eq. (C1a)shows that a must satisfy the linearized equation at O(ε),

κ∂2a

∂ζ2± i√

1 + 4κβcw∂a

∂ζ+ iα

∂a

∂τ+s

2

∂2a

∂τ2

+ ρ0f′(ρ0) (a+ a∗) ' 0.

(C4)

Following the method detailed in Ref. [11], the Fouriermodes of Eq. (C4) have a complex propagation constantKp (allowing for potential growth of the perturbationwave) and frequency Ωp that are connected by the quarticdispersion relation

κ2K4p −

[1 + 4κβcw + 2κρ0f

′(ρ0)− sκΩ2p

]K2

p

∓ 2αΩp

√1 + 4κβcwKp

+(

12Ω2

p

) [(12Ω2

p

)− 2α2 − 2sρ0f

′(ρ0)]

= 0. (C5)

While the four roots of Eq. (C5) can be written downexactly [51], they are algebraically cumbersome and donot provide much further physical insight. However, oneregime of key importance of that corresponding to long-wavelength excitations.

18

2. Long-wavelength instabilities

Neglecting all κ-dependent contributions to the MIspectrum can be expected to recover the long-wavelengthresult from conventional pulse theory. By considering aforward-travelling host wave, perturbation dispersion re-lation (C5) is well-described by the quadratic approxi-mation

K2p + 2αΩpK −

(12Ω2

p

) [(12Ω2

p

)− 2α2 − 2sρ0f

′(ρ0)]' 0

(C6a)

which has the two solution branches

Kp = −αΩp ±√(

12Ω2

p

) [(12Ω2

p

)− 2sρ0f ′(ρ0)

]. (C6b)

The term at −αΩp is transformed away after a Galileanboost to the local time-frame, leaving the familiar resultfor MI in NLS-based models. One can thus deduce thatthe long-wavelength MI properties of the system are es-sentially independent of κ, and that they occur whenever2sρ0f

′(ρ0) > Ω2p/2.

Our principal interest lies with model (C1b), wheref ′(ρ0) ≥ 0 for ρ0 ≥ 0. In the anomalous-GVD regime(where s = +1), it follows that MI appears in the long-wave band 0 < Ω2

p < 4ρ0f′(ρ0). The most unsta-

ble frequency Ωp0 is then easily calculated from Ω2p0 =

2ρ0f′(ρ0). In contrast, Eqs. (C6a) and (C6b) predict

that there is no long-wavelength MI in the normal-GVDregime (where s = −1). Hence, the background wave ofdark soliton (28a) is expected to be always stable againstsmall background disturbances.

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